Kardar-Parisi-Zhang Universality Class, Integrable Differential Equations, and Spin Glass

This project is about how sometimes complex and random interactions between the constituents in a large probabilistic system may lead to simple and universal behaviors. Here universality means that these properties are attained for a large class of such systems. A few concrete classes of probabilistic models and their fundamental properties will be investigated, especially when the system sizes become large. This project will help improve our understanding of the scopes and limitations of the universality of probability models. Many parts of the project will be carried out with graduate students and young researchers, helping their professional development. In concrete terms, the Kardar-Parisi-Zhang (KPZ) universality class models will be studied as well as their connections to integrable differential equations, and spin glass models. Of particular interest will be the multi-time distributions of KPZ universality class models on the ring domain, to understand and broaden the relations between the KPZ models and integrable differential equations, and to study the fluctuations of the spherical Sherrington-Kirkpatrick model perturbed by a deterministic field. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.